This story was told to me by one of my acquaintances long time ago. It touched my soul so deeply that I have remembered it for years. Moreover, this story had strongly changed my outlook. This is what I am going to tell you about right now. Let me begin with the story itself. It happened in the times of the Soviet Empire. Though I cannot tell you the name of my acquaintance and what train crash could have happened and when, I can tell you for sure that this is a true story. So, let me present to your attention a play in three acts with epilogue and prologue.

Prologue and Background

In some sense the trip could decide the fate of my acquaintance. It was supposed to be a long railway trip. As usual, my acquaintance arrived at the station far before the train’s departure, made himself comfortable in the compartment and started to read a newspaper. “Then suddenly my stomach gurgled and a nightmare began – I felt terrible gripes”. As you surely know, the lavatory in Soviet trains was closed at stops. What should he do? His neighbor advised him to visit the lavatory at the station, as there were still fifteen minutes left before the departure. “Can you imagine, I felt so bad that I had to use his advice”. Though he assured me that he had had aches in his stomach before, still he had never felt so bad. “Then my watch stopped as ill lick would have it. I noticed it only at the platform. You know, a second-hand in my watch was so thin that I had never paid attention to it. Looking at my watch I was happy at having so much time left before the departure.” When my friend came back to the platform, his nightmare went on – the train had already left with the luggage and all the documents. “It was the moment when I noticed my watch stop. I got so angry that I smashed them into pieces, though it was my good friend’s present”. Then he surely turned for help to the railway administration and police. But there was nothing they could do unless running after the train. “The most interesting thing is that the pain disappeared immediately after I saw that the train had left. Later I thought that it had happened because of fear.” Upset, my friend had to come back home to make his trip next time. “My hair rose when I learned that there had been a terrible train crash and no one from my van survived. Then I thought that I would have placed the watch, which was already in the dump, under the icons if I could find it”. From that very moment on my friend became a very faithful person and his life really changed for the better.

Act 1 or General Reasonings

“Mysticism” was my first thought, while some people would suppose it was a good luck. Is it a fate or is there anything more about that? To analyze this story I applied my experience in system analysis (a system approach to things and events). Finally I came to the conclusion that everything has its reason.

Of course, from the point of view of an ordinary observer, it was just a coincidence, a very lucky chance, which saved the life of my friend. But let us imagine a ball rolling down the gutter. With a certain probability we can foresee the ball’s position in the near future – where it will be and when; applying a simple formula from the course of general physics we can guess the moment when it falls down from the gutter. The main conclusion from this experiment is that both the ball and the gutter compose a mutual system, isolated in a sense.

Using the same line of reasoning like in the physical experiment, we come to the conclusion that my friend did not belong to the system of the crashed train, but he played a certain role in another system where he was not supposed to perish (as his system was far more stable).

Now let us imagine that we can predict the fate of any system and we can choose what system to belong to. Thus, we can control our future in a sense. If my friend had known what would have happened he would have put a monument to his watch.

So, the main conclusion we can draw from this story is that we are all screws, constituent parts of a system functioning and developing according to its own rules. Thus, being aware of all the secrets of our system, we can predict the future. And the influence of the system as well as our influence on the system can be of different kinds (physical, chemical, biological, etc.).

Act 2 or Drawing Schemes

The scheme demonstrates a hypothetical ratio between arbitrary systems. The intersection of their boundaries is just my supposition. One should consider that the boundaries may not intersect but at the same time the systems may interact, and that the boundaries may be of any kind (geographic, demographic, financial, and political).

The scheme shows that if an object under analysis is located in the intersection area, it is attracted by the centers of several systems, though it belongs just to one of them. Belonging to one system is just a temporary factor, which depends on any influence redistribution among the systems. Thus, in the near future the object can be expected to move to another system.

By analogy, the same happened to my friend: he did not enter the system of the train, though the attraction was very strong (he had to make his trip by train). Obviously, he was used (or attracted) by another system, so he was not to leave by that train.

It is well-known that systems can apply various means of influence to cooperate with their elements. It can be physical, physiological as well as any other type of influence. For example, previously I had serious doubts regarding the fact that icons could help win battles (that was quite often in the Ancient Russia). However, now I have changed my view on this problem. There can be no trifles in the life of system and its elements, so an icon in the battlefield could strengthen the impact of center (the faith) on the final elements (the soldiers), that raised the morale of the army and finally lead to victory.

Act 3 or Predicting System’s Condition

A ball rolling down the gutter is the simplest system. We can predict its position at any moment in the future, as we obtain all the information on the considered phenomenon. Now let us imagine a break in the gutter the ball can fall down through. For a system where the ball is supposed to constantly roll down, such break can be a disaster especially if the ball is flimsy (made of thin glass, for example), as it can fall into pieces. So, we have just predicted the ball’s crash (an important event) with a high probability.

In reality we all belong to some systems, which lead us to falls, rises or something else. In order to foresee the future we just have to see the whole of a system (systems) we cooperate with now or will cooperate with in the future and the way this system (systems) is (are) likely to function in the future. You may say that it is almost a mystery. But in fact, to be able to see and guess the consequences of some events, it is enough to be a good observer and analytic.

To my opinion, a phenomenon of famous foretellers does not exist, but there are just gifted people able to see the general course of things and to predict its future behavior. If a gipsy fortuneteller has such a gift, you can rely on her without any doubt. Various attributes of fortunetelling like playing-cards or stars are just means of building correlative patterns of our system on the time axis. And the more accurate the correspondence between the system’s behavior scale (here a ruler and a set square) and the physical system (the ball and the gutter), the more precise the fortunetelling.

So, is it possible to control the future? It is you to decide, as some things are actually under our control, but others are not. It depends on the strength of the system’s influence and on how we can control the latter. Imagine, that you have foreseen that your system is near to a catastrophe, so why don’t you move to another system (see the scheme above)? If you feel that a plain is likely to crash, then maybe you should better stay at the airport (or not go to work or to your friends’ home). If we set a magnet near the break in the gutter, the ball will get magnetized and will not fall into pieces.

Epilogue or Ode to Rarefied Sets

Now let us discuss the foretelling with the help of scientific approach. Here we will apply the notion of probability (its Majesty the probability).

The picture represents a possible variant consisting of two systems, one of them supposed to be more preferable (perfect career, good health or saving life, etc.). The picture shows that we are being influenced by various systems. Thus, we have to estimate the development trends of these systems as well as the probability of our presence in any of them.

We cannot determine strict boundaries between the fields, so each field is probabilistic. At the same time we can estimate the risk of our belonging to any system and, what is more important, we can guess the probability of our participation in some event. Everything here depends on the precise description of both the system and its development, as well as on the observer’s abilities, the means he uses, and on whether the development pattern of the considered phenomenon corresponds to reality. (Indeed, in physics experiments on correspondence of mathematic patterns to real physical processes play a very important role).

I have divided these probabilities (probability of belonging and probability of participation) intentionally. Actually, the factors determining our position relative to systems “A” and “B”, are quite different. This is the essence of rarified sets, which are widely represented in one of my articles. The mathematical machine, that I proposed, helps analyze our position relative to quite different systems at the level of probabilistic descriptions. This is the method the fortunetellers apply, as they just need a probabilistic means somehow correlated with our fate (!). Further they just have to predict which events you will participate in and which not. However, this process requires detached substances (“queens”, “aces”, “kings” and other cards correspond to certain patterns of behavior — love, health, traveling, etc.). The preciseness of correlation between the cards sequence (stars arrangement) and our experience (character, problem) depends on the degree of competence of the observer (the fortuneteller).

Any system we deal with always (!) sends us some signals of this or that form. My personal experience proves the idea, and I am convinced that you can also draw certain situations from your life when a system sent you alarm signals and you just neglected them and got into trouble. My pictures also illustrate a certain power that influences us, which means that we can feel it. The latter proves the idea that we can predict our own future. To do that we just have to see the essence of systems and feel their influence (like Nostradamus). We can also try to describe our future through some mathematical patterns. Anyways, we are likely to get one and the same result — “together with system”, which means that one should not swim up-stream, but he should gain profit from any situation so that being in any system he could choose the best position both inside this system and relative to other systems.

Here I would like to draw your attention to the idea of independence of two possibilities: whether an element belongs to a certain set or not. This idea occurred to me while trying to understand the nature of our dreams. Walking straight by the road you will face certain events and environment, while if you walk in the opposite direction, events and environment – everything will be quite different from what you have seen before. So, the two roads (straight and back) in your dreams are a kind of an asymmetric world represented to us as increased and dynamic.

On the one hand, if we try to analyze whether an element possibly belongs to a certain multitude we will apply specific notions. While on the other, if we try to prove that the element does not belong to this multitude we are likely to use other indexes.

Certainly, all the analyzed factors have much in common. However, this interrelation is rather complicated and implicit. (Let us suppose that they are multifractals, and that we need to establish the interrelation between their primings, which we do not know). The trick is that at a subconscious level we are able to connect, apparently untied factors, while at a conscious level we have not yet learned to do that.

For this case we have a dream machine. So, I have chosen the way of dream formation – having broken the connection between what can be with something that cannot be. Let me draw an example with dreams. Imagine, that today you have seen a beautiful building, which has seized your attention. Accordingly, it is highly possible that today in your night dream you will see some constructions. In the evening, watching news on TV, you see unpleasant staff of some international conflict. So, the possibility of seeing constructions in your dream still remains, while the possibility of not seeing them arises, as in your dream you may experience some kind of violence, etc. As a result, if moving straight in your dream you will come across some buildings, but try walking back and you will meet the war. Does not it remind you a compromise between our view of the world and our representation of the world?

Further on, trying to find the message of our dreams, I found out that we deal with interpretation of our problems in dreams in some aspects, which we had not noticed before when being awake. These aspects, however, are able to influence decision-making on certain problems. Is it worth moving straight if you will never get back. Then I tried to find linkage between my vision of the world in dreams and mathematics, and consequently between the notion of problem and some mathematical instrument.

It is not a secret that I am an admirer of mathematical machine of indistinct sets (fuzzy logic). Naturally, I either apply or plan to apply this instrument in all my projects. However, I often come across the problems of presence and absence of element in certain sets (actually the road back). Certainly you may assume that having calculated the probability of presence of element in the set I can easily estimate the probability of its absence in the set. Here that I supposed that these two probabilities may be absolutely different and have nothing in common. Moreover, the presence of element in the set and its absence get interrelated if we apply the same factors for analysis. While, actually it is just theory, and in practice factors influencing the presence or absence of element in a set are quite different.

Thus, the rarefied set can be described as follows: {e1(p1,z1),e2(p2,z2),e3(p3,z3)...en(pn,zn)} (the rarefied set), with “p” as probability of presence of “e” element in the set, and “z” as probability of its absence. I suppose, that the notion of «problem» can be well described with the help of such mathematical means. For this purpose let us analyze the following diagram.

The diagram shows the variations of uncertainty degree of analyzed sets according to the transition between the following notions: “problem — purpose — task”. One degree of probability disappears with each notion starting with the notion of problem! When analyzing problems we should consider the probabilities of fulfillment and non-fulfillment, while when analyzing purposes and tasks we should consider probability of achievement and a set of measures respectively.

The notion of risk is another idea, which requires some judgment. So I am not able to make any statements on this matter yet. However, I suppose that risk is somehow related to the probabilities of presence and absence of an element in a set.

The next step in development of idea should be the creation of some tool permitting to apply the idea in practice (something like Gant’s diagrams). The diagram includes the following marks: “M” as a set describing this or that notion, “e” as an element of a set (the factor under analysis), “p” as probability of presence of element in the set, “z” as probability of its absence in the set (here “p” and “z” are independent figures). In case with “PROBLEM” notion we deal with a rarefied set, where the two probabilities of presence and absence do not depend on each other. I think that mathematical means of complex figures could be applied here. However, it is just my supposition.

Let me draw another example fro the military field. Various factors influence probability of presence and absence of enemy forces in some specific place. For example, their presence is highly possible due to the tactic expedience. While, some climatic factors can influence the probability of their absence (World War II has shown that in Russia, for example, tanks are almost no use in rainy weather and in swampy areas). I mean that weather hardly depend on the battlefield situation.
Certainly, the idea on the rarefied sets still requires deep analysis from the point of view of the tool itself and its usage. However, its development and use in various fields of knowledge seems to be very expedient. It is enough to look at the diagram to see the variations of uncertainty degree of analyzed sets according to the transition between the following notions: “problem — purpose — task”.